Colour an element of white if its coordinates are coprime and black otherwise. What does this colouring look like when seen from a “uniformly chosen” random point of ? More generally, label every element of by its greatest common divisor: what do the labels look like around a “uniform” random point of ? We answer these questions and generalisations of them, which includes a result a “local/graphon” convergence. We also investigate the percolative properties of the colouring under study.
I acknowledge the support of the ERC Advanced Grant 740943 GeoBrown.
At the end of a talk given by Nathanaël Enriquez, Bálint Virág asked him about the local limit of visible lattice points: I would like to thank Bálint Virág for asking this question and Nathanaël Enriquez for letting me know of it. I am also grateful to Nathanaël Enriquez for many enthusiastic discussions about this project. I am thankful to Subhajit Goswami and Aran Raoufi for comments on an earlier version of this paper, to Samuel Le Fourn for bibliographical help, and to the anonymous referees for their feedback. I am grateful to my former postdoctoral advisors Nicolas Curien and Jean-François Le Gall, as well as the Université Paris-Sud, for having provided me with an excellent working environment.
"On coprime percolation, the visibility graphon, and the local limit of the GCD profile." Electron. Commun. Probab. 27 1 - 14, 2022. https://doi.org/10.1214/21-ECP381